Quadratic numerical semigroups of embedding dimension five: two questions of Stamate
Abstract
Let H be a numerical semigroup, K[H] its semigroup ring and grₘ K[H] its tangent cone; H is called quadratic when the defining ideal I^(*)_(H) of grₘ K[H] is generated by quadrics. Stamate gave a maximal list of twelve h-vectors for a quadratic H of embedding dimension five, exhibited an example for each except (1,4,5), and asked whether the list should be reduced further; in the same paper he isolated three artinian ideals J₁,J₂,J₃ of K[x₂,x₃,x₄,x₅] arising in his screening, reported that he could find no quadratic numerical semigroup producing them, and observed that if none existed the hypotheses "K algebraically closed, charK ≠ 2" could be removed from his theorem that Koszul and G-quadratic agree in embedding dimension five. We settle both questions. No numerical semigroup of embedding dimension five whose tangent cone has h-vector (1,4,5) is quadratic, so the list of twelve h-vectors becomes a list of eleven and the missing entry stays missing. The second question has a mixed answer: J₂ is produced by no quadratic numerical semigroup, for any labelling of the variables, whereas J₁ is produced by ⟨ 7,23,31,39,40⟩ and J₃ by ⟨ 8,26,35,44,45⟩, both quadratic — so the hypothesis under which the assumptions on K were to be dropped is false. The proofs turn each question into finite data: the Cohen–Macaulayness of the tangent cone is derived from the Hilbert function rather than assumed, the degree-three condition imposed by quadratic generation becomes a connectivity question on twenty vertices, and what survives is killed by a congruence obstruction on Apéry sets. All finite searches are machine-checked in Lean 4.
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Stated results
Q1candidate2026-09-03
The ten Apery values forced by the surviving (1,4,5) configuration are never pairwise distinct mod 10; nine residues are attainable, so the bound is sharp
Q2candidate2026-09-03
Over N: if a2+a5 = a3+a4 then 0, a2..a5 and the five sums a2+a3, a2+a4, a2+a5, a3+a5, a4+a5 are not pairwise incongruent mod 10, so they are not an Apery set Ap(H,10)
Q3candidate2026-09-03
Finite classification: over all nine realizable coincidence patterns and all 2¹0 star-sets, the only degree-3-consistent configuration with h2 = 5 is a2+a5 = a3+a4 with star exactly the four squares, i.e. J = (x2², x3², x4², x5², x2x5 - x3x4)
Q4candidate2026-09-03
No numerical semigroup with embdim 5 whose tangent cone has h-vector (1,4,5) is quadratic – Stamate's Table 1 reduces from twelve h-vectors to eleven
Q5candidate2026-09-03
The eight Apery values forced by Stamate's ideal J2 are never pairwise distinct mod 8; seven residues are attainable
Q6candidate2026-09-03
No quadratic numerical semigroup with embdim 5 produces Stamate's ideal J2, for any labelling of the variables
Q7known data2026-09-03
All twelve h-vectors of Table 1 of arXiv:1512.04893v3 recomputed in Lean from the order function, and all twelve rows confirmed quadratic (no minimal generator of I*_H in degrees 3-5)
Q8candidate2026-09-03
<7,23,31,39,40> and <8,26,35,44,45> are quadratic numerical semigroups producing Stamate's ideals J1 and J3 exactly – the examples the source could not find
Q9routine2026-09-03
The eight smallest numerical semigroups with h-vector (1,4,5) each acquire a minimal generator of I*_H in degree 3, hence are not quadratic
Provenance
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- Machina Mathematica
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- Source context
- Source. Dumitru I. Stamate, *On the Cohen–Macaulay property for quadratic tangent cones*, arXiv:1512.04893 (v3, 2016-08-09); *Electron. J. Combin.* 23 (2016), Paper 3.20, DOI 10.37236/5793. Companion: J. Herzog, D. I. Stamate, *Quadratic numerical semigroups and the Koszul property*, arXiv:1510.00935, Kyoto J. Math.
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